Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 14.9 → 5.7

Time: 13.6s

Precision: binary64

Math

\[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

↓

\[\begin{array}{l} t_1 := t + \left(\left(\frac{y}{z} \cdot x + \frac{a}{\frac{z}{t}}\right) - \left(\frac{y}{\frac{z}{t}} + x \cdot \frac{a}{z}\right)\right)\\ t_2 := \frac{a - z}{z}\\ t_3 := \left(\left(x + \frac{x}{t_2}\right) + \frac{y}{a - z} \cdot \left(t - x\right)\right) - z \cdot \frac{t}{a - z}\\ \mathbf{if}\;z \leq -3.682259159630762 \cdot 10^{+209}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.9658125341391284 \cdot 10^{-195}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 7.326205365430645 \cdot 10^{-52}:\\ \;\;\;\;x + \left(\frac{t - x}{\frac{a - z}{y}} - \frac{t}{t_2}\right)\\ \mathbf{elif}\;z \leq 2.6384624238560835 \cdot 10^{+141}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (* (- y z) (/ (- t x) (- a z)))))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1
         (+
          t
          (- (+ (* (/ y z) x) (/ a (/ z t))) (+ (/ y (/ z t)) (* x (/ a z))))))
        (t_2 (/ (- a z) z))
        (t_3
         (-
          (+ (+ x (/ x t_2)) (* (/ y (- a z)) (- t x)))
          (* z (/ t (- a z))))))
   (if (<= z -3.682259159630762e+209)
     t_1
     (if (<= z -3.9658125341391284e-195)
       t_3
       (if (<= z 7.326205365430645e-52)
         (+ x (- (/ (- t x) (/ (- a z) y)) (/ t t_2)))
         (if (<= z 2.6384624238560835e+141) t_3 t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	return x + ((y - z) * ((t - x) / (a - z)));
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = t + ((((y / z) * x) + (a / (z / t))) - ((y / (z / t)) + (x * (a / z))));
	double t_2 = (a - z) / z;
	double t_3 = ((x + (x / t_2)) + ((y / (a - z)) * (t - x))) - (z * (t / (a - z)));
	double tmp;
	if (z <= -3.682259159630762e+209) {
		tmp = t_1;
	} else if (z <= -3.9658125341391284e-195) {
		tmp = t_3;
	} else if (z <= 7.326205365430645e-52) {
		tmp = x + (((t - x) / ((a - z) / y)) - (t / t_2));
	} else if (z <= 2.6384624238560835e+141) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + ((y - z) * ((t - x) / (a - z)))
end function

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = t + ((((y / z) * x) + (a / (z / t))) - ((y / (z / t)) + (x * (a / z))))
    t_2 = (a - z) / z
    t_3 = ((x + (x / t_2)) + ((y / (a - z)) * (t - x))) - (z * (t / (a - z)))
    if (z <= (-3.682259159630762d+209)) then
        tmp = t_1
    else if (z <= (-3.9658125341391284d-195)) then
        tmp = t_3
    else if (z <= 7.326205365430645d-52) then
        tmp = x + (((t - x) / ((a - z) / y)) - (t / t_2))
    else if (z <= 2.6384624238560835d+141) then
        tmp = t_3
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + ((y - z) * ((t - x) / (a - z)));
}

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t + ((((y / z) * x) + (a / (z / t))) - ((y / (z / t)) + (x * (a / z))));
	double t_2 = (a - z) / z;
	double t_3 = ((x + (x / t_2)) + ((y / (a - z)) * (t - x))) - (z * (t / (a - z)));
	double tmp;
	if (z <= -3.682259159630762e+209) {
		tmp = t_1;
	} else if (z <= -3.9658125341391284e-195) {
		tmp = t_3;
	} else if (z <= 7.326205365430645e-52) {
		tmp = x + (((t - x) / ((a - z) / y)) - (t / t_2));
	} else if (z <= 2.6384624238560835e+141) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	return x + ((y - z) * ((t - x) / (a - z)))

↓

def code(x, y, z, t, a):
	t_1 = t + ((((y / z) * x) + (a / (z / t))) - ((y / (z / t)) + (x * (a / z))))
	t_2 = (a - z) / z
	t_3 = ((x + (x / t_2)) + ((y / (a - z)) * (t - x))) - (z * (t / (a - z)))
	tmp = 0
	if z <= -3.682259159630762e+209:
		tmp = t_1
	elif z <= -3.9658125341391284e-195:
		tmp = t_3
	elif z <= 7.326205365430645e-52:
		tmp = x + (((t - x) / ((a - z) / y)) - (t / t_2))
	elif z <= 2.6384624238560835e+141:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
end

↓

function code(x, y, z, t, a)
	t_1 = Float64(t + Float64(Float64(Float64(Float64(y / z) * x) + Float64(a / Float64(z / t))) - Float64(Float64(y / Float64(z / t)) + Float64(x * Float64(a / z)))))
	t_2 = Float64(Float64(a - z) / z)
	t_3 = Float64(Float64(Float64(x + Float64(x / t_2)) + Float64(Float64(y / Float64(a - z)) * Float64(t - x))) - Float64(z * Float64(t / Float64(a - z))))
	tmp = 0.0
	if (z <= -3.682259159630762e+209)
		tmp = t_1;
	elseif (z <= -3.9658125341391284e-195)
		tmp = t_3;
	elseif (z <= 7.326205365430645e-52)
		tmp = Float64(x + Float64(Float64(Float64(t - x) / Float64(Float64(a - z) / y)) - Float64(t / t_2)));
	elseif (z <= 2.6384624238560835e+141)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + ((y - z) * ((t - x) / (a - z)));
end

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = t + ((((y / z) * x) + (a / (z / t))) - ((y / (z / t)) + (x * (a / z))));
	t_2 = (a - z) / z;
	t_3 = ((x + (x / t_2)) + ((y / (a - z)) * (t - x))) - (z * (t / (a - z)));
	tmp = 0.0;
	if (z <= -3.682259159630762e+209)
		tmp = t_1;
	elseif (z <= -3.9658125341391284e-195)
		tmp = t_3;
	elseif (z <= 7.326205365430645e-52)
		tmp = x + (((t - x) / ((a - z) / y)) - (t / t_2));
	elseif (z <= 2.6384624238560835e+141)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t + N[(N[(N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision] + N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y / N[(z / t), $MachinePrecision]), $MachinePrecision] + N[(x * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - z), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x + N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.682259159630762e+209], t$95$1, If[LessEqual[z, -3.9658125341391284e-195], t$95$3, If[LessEqual[z, 7.326205365430645e-52], N[(x + N[(N[(N[(t - x), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - N[(t / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.6384624238560835e+141], t$95$3, t$95$1]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

1. Split input into 3 regimes

2. if z < -3.68225915963076195e209 or 2.6384624238560835e141 < z

   1. Initial program 28.1

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around inf 23.8

      \[\leadsto \color{blue}{\left(\frac{y \cdot x}{z} + \left(t + \frac{a \cdot t}{z}\right)\right) - \left(\frac{y \cdot t}{z} + \frac{a \cdot x}{z}\right)} \]

   3. Simplified 8.2

      \[\leadsto \color{blue}{t + \left(\left(\frac{y}{z} \cdot x + \frac{a}{\frac{z}{t}}\right) - \left(\frac{y}{\frac{z}{t}} + \frac{a}{z} \cdot x\right)\right)} \]

   if -3.68225915963076195e209 < z < -3.96581253413912841e-195 or 7.3262053654306445e-52 < z < 2.6384624238560835e141

   1. Initial program 12.5

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Applied egg-rr 12.5

      \[\leadsto x + \color{blue}{\left(y \cdot \frac{t - x}{a - z} + \left(-z\right) \cdot \frac{t - x}{a - z}\right)} \]

   3. Taylor expanded in x around 0 17.8

      \[\leadsto \color{blue}{\left(\frac{z \cdot x}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(\frac{y \cdot x}{a - z} + \frac{t \cdot z}{a - z}\right)} \]

   4. Simplified 5.6

      \[\leadsto \color{blue}{\left(\left(x + \frac{x}{\frac{a - z}{z}}\right) + \frac{y}{a - z} \cdot \left(t - x\right)\right) - \frac{t}{a - z} \cdot z} \]

   if -3.96581253413912841e-195 < z < 7.3262053654306445e-52

   1. Initial program 7.0

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Applied egg-rr 7.0

      \[\leadsto x + \color{blue}{\left(y \cdot \frac{t - x}{a - z} + \left(-z\right) \cdot \frac{t - x}{a - z}\right)} \]

   3. Taylor expanded in t around inf 7.5

      \[\leadsto x + \left(y \cdot \frac{t - x}{a - z} + \color{blue}{-1 \cdot \frac{t \cdot z}{a - z}}\right) \]

   4. Simplified 7.6

      \[\leadsto x + \left(y \cdot \frac{t - x}{a - z} + \color{blue}{\frac{-t}{\frac{a - z}{z}}}\right) \]

   5. Taylor expanded in y around 0 8.0

      \[\leadsto x + \left(\color{blue}{\frac{y \cdot \left(t - x\right)}{a - z}} + \frac{-t}{\frac{a - z}{z}}\right) \]

   6. Simplified 3.7

      \[\leadsto x + \left(\color{blue}{\frac{y}{a - z} \cdot \left(t - x\right)} + \frac{-t}{\frac{a - z}{z}}\right) \]

   7. Applied egg-rr 3.6

      \[\leadsto x + \left(\color{blue}{\frac{t - x}{\frac{a - z}{y}}} + \frac{-t}{\frac{a - z}{z}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.682259159630762 \cdot 10^{+209}:\\ \;\;\;\;t + \left(\left(\frac{y}{z} \cdot x + \frac{a}{\frac{z}{t}}\right) - \left(\frac{y}{\frac{z}{t}} + x \cdot \frac{a}{z}\right)\right)\\ \mathbf{elif}\;z \leq -3.9658125341391284 \cdot 10^{-195}:\\ \;\;\;\;\left(\left(x + \frac{x}{\frac{a - z}{z}}\right) + \frac{y}{a - z} \cdot \left(t - x\right)\right) - z \cdot \frac{t}{a - z}\\ \mathbf{elif}\;z \leq 7.326205365430645 \cdot 10^{-52}:\\ \;\;\;\;x + \left(\frac{t - x}{\frac{a - z}{y}} - \frac{t}{\frac{a - z}{z}}\right)\\ \mathbf{elif}\;z \leq 2.6384624238560835 \cdot 10^{+141}:\\ \;\;\;\;\left(\left(x + \frac{x}{\frac{a - z}{z}}\right) + \frac{y}{a - z} \cdot \left(t - x\right)\right) - z \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t + \left(\left(\frac{y}{z} \cdot x + \frac{a}{\frac{z}{t}}\right) - \left(\frac{y}{\frac{z}{t}} + x \cdot \frac{a}{z}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))